Corrigendum to “Inference on impulse
response functions in structural VAR models”
[J. Econometrics 177 (2013), 1-13]
Atsushi Inouea Lutz Kilianb∗
a
b
Department of Economics, Vanderbilt University, Nashville TN 37235-1819
University of Michigan, Department of Economics, Ann Arbor, MI 48109-1220
Proposition 1 in Inoue and Kilian (2013) for the posterior density of the
set of structural impulse responses in the fully sign-identified VAR model is
not correct as stated. The correct statement and proof is provided below.
The correction does not affect the substance of the empirical findings based on
this proposition. The statement and proof of Proposition 2 and the empirical
results based on that proposition are not affected.
Consider an n-dimensional VAR(p) model with an intercept. Ignoring the
intercept for notational convenience, let B = [B1 · · · Bp ]0 denote the slope
parameters of the model. Let Σ denote the error covariance matrix. Let A
be the lower-triangular Cholesky decomposition of Σ such that AA0 = Σ, and
let vech(A) denote the n(n + 1)/2 × 1 vector that consists of the on-diagonal
elements and the below-diagonal elements of A. We assume that the n × n
real rotation matrix U is orthogonal, U 0 U = In , and that its determinant is
|U | = 1. Then
S = In − 2(In + U )−1
(1)
is an n × n skew-symmetric matrix (León et al. (2006), p. 414). Let s denote
the n(n − 1)/2 × 1 vector that consists of the below-diagonal elements of
S. Then U and s have a one-to-one relationship. When U is uniformly
distributed over the space of n × n real matrices such that U 0 U = In and
∗∗
Tel: +1-734-647-5612. E-mail: [email protected].
1
|U | = 1, the density of s is given by
f (s) =
Γ(i/2)
Πni=2 i/2
π
!
2(n−1)(n−2)/2
|In + S|n−1
(2)
(see equation 4 of León et al. (2006), p. 415).1
Let Φ = [Φ01 Φ02 · · · Φ0p ]0 where Φi is the ith reduced-form vector
moving average coefficient matrix. There is a one-to-one mapping between
the first p + 1 structural impulse responses Θ̃ = [A, Φ1 A, Φ2 A, · · · , Φp A],
where A = AU , on the one hand, and the tuple formed by the reduced-form
VAR parameters and s, (B, vech(A), s), on the other. The nonlinear function
e = h(B, vech(A), s) is known. Using the change-of-variables method, the
Θ
e can be written as
posterior density f of Θ
f (Θ̃) =
=
∝
∂[vec(B)0 vech(A)0 s0 ] f (B, vech(A), s)
∂vec(Θ̃)
−1 ∂vech(Σ) ∂vec(Θ̃)
f (B, Σ, s)
∂[vec(B)0 vech(A)0 s0 ] ∂vech(A)0 −1 ∂vech(Σ) ∂vec(Θ̃)
f (B|Σ)f (Σ)f (s),
∂[vec(B)0 vech(A)0 s0 ] ∂vech(A)0 (3)
where B, Σ = AA0 , and U are the unique values that satisfy the nonlinear
e = h(B, vech(A), s). The conditioning on the data is omitted for
function Θ
notational simplicity.
Let Dn denote the n2 × n(n + 1)/2 duplication matrix of zeros and ones
such that vec(M ) = Dn vech(M ) for any n × n symmetric matrix M (see
Definition 4.1 of Magnus (1988), p. 55). Dn+ denotes the Moore-Penrose
inverse of Dn , i.e., Dn+ = (Dn0 Dn )−1 Dn0 , so that we can write vech(M ) =
Dn+ vec(M ) (see Theorem 4.1 of Magnus (1988), p. 56). Let Ln denote the
n(n + 1)/2 × n2 elimination matrix of zeros and ones such that vec(M ) =
L0n vech(M ) for any lower triangular matrix M (see Definition 5.1 of Magnus
(1988), p. 76). Kn denotes the n2 × n2 communication matrix such that
1
León et al. (2006, p. 416) define s as the vector that consists of the above-diagonal
elements of S. Let sleon denote this vector. Because S is skew-symmetric, there is a
n(n − 1)/2 × n(n − 1)/2 permutation matrix P such that sleon = −P s. Because the
absolute value of the determinant of −P is always 1, the density remains the same as in
(2) even when s is defined to be the vector obtained from the below-diagonal elements of
S.
2
vec(M 0 ) = Kn vec(M ) for any n × n matrix M (see Magnus and Neudecker
(1999), pp. 46–47). D̃n is the n2 × n(n − 1)/2 matrix such that vec(S) = D̃n s
(see Definition 6.1 in Magnus (1988), p. 94).
Proposition 1. The posterior density of Θ̃ is
f (Θ̃) ∝
−1 ∂vech(Σ) ∂vec(Θ̃)
f (B|Σ)f (Σ)f (s)
∂[vec(B)0 vech(A)0 s0 ] ∂vech(A)0 −1
= (|(U 0 ⊗ In )L0n (In ⊗ A)JU | |A|np )
× Dn+ [(A ⊗ In ) + (In
⊗ A)Kn ]L0n f (B|Σ)f (Σ)f (s)
(4)
where JU = 2[(In − S0 )−1 ⊗ (In − S)−1 ]D̃n .
Derivation of the Jacobian matrix in Proposition 1: Because U = 2(In −
S)−1 − In ,
dU = 2(In − S)−1 (dS)(In − S)−1 ,
(5)
and thus
vec(dU ) = 2[(In − S)−10 ⊗ (In − S)−1 ]vec(dS)
= 2[(In − S)−10 ⊗ (In − S)−1 ]D̃n ds
= JU ds.
(6)
Because
d(AU ) = (dA)U + A(dU ) = (dA)U + 2A(In − S)−1 (dS)(In − S)−1 ,(7)
d(ΦAU ) = (dΦ)AU + Φ(dA)U + ΦA(dU )
= (dΦ)AU + Φ(dA)U + 2ΦA(In − S)−1 (dS)(In − S)−1 ,
(8)
it follows that
(U 0 ⊗ In )dvec(A) + 2(In ⊗ A)((In − S0 )−1 ⊗ (In − S)−1 )dvec(S),
(U 0 ⊗ In )L0n dvech(A) + 2(In ⊗ A)((In − S0 )−1 ⊗ (In − S)−1 )D̃n ds,
(U 0 ⊗ In )L0n dvech(A) + 2(In ⊗ A)JU ds,
(9)
0 0
0
(U A ⊗ Inp )dvec(Φ) + (U ⊗ Φ)dvec(A)
+2(In ⊗ ΦA)((In − S0 )−1 ⊗ (In − S)−1 )dvec(S)
= (U 0 A0 ⊗ Inp )dvec(Φ) + (U 0 ⊗ Φ)L0n dvech(A) + (In ⊗ ΦA)JU ds.(10)
dvec(AU ) =
=
=
dvec(ΦAU ) =
3
It follows from expressions (9) and (10) that
#
"
On2 ×n2 p
(U 0 ⊗ In )L0n (In ⊗ A)JU
.
(U 0 ⊗ Φ)L0n (In ⊗ ΦA)JU U 0 A0 ⊗ Inp
(11)
0
0
2
2
Since the upper-left submatrix, [(U ⊗ In )Ln (In ⊗ A)JU ], is n × n , the
upper-right submatrix is the n2 × n2 p submatrix of zeros, and the lower-right
submatrix, U 0 A0 ⊗ Inp , is n2 p × n2 p, (11) is block lower triangular. Thus its
determinant is given by the product of determinants:
∂vec(Θ̃)
J1 ≡
=
∂[vech(A)0 s0 vec(Φ)0 ]
|J1 | = |(U 0 ⊗ In )L0n (In ⊗ A)JU | |U 0 A0 ⊗ Inp |
= |(U 0 ⊗ In )L0n (In ⊗ A)JU | |A|np .
(12)
Because of the recursive relationship defined by equations (11), (14) and (17)
in Inoue and Kilian (2013), the Jacobian matrix of θ with respect to B is
block-diagonal and each diagonal block has unit determinant. Thus
|J2 | ≡
∂vec(Φ) ∂vec(B) = 1.
(13)
Since the Jacobian of vec(Σ) with respect to vec(A) is
[(A ⊗ In ) + (In ⊗ A)Kn ],
(14)
the determinant of the Jacobian of vech(Σ) with respect to vech(A) is given
by
|J3 | ≡ Dn+ [(A ⊗ In ) + (In ⊗ A)Kn ]L0n (15)
Thus, Proposition 1 follows from expressions (12), (13) and (15).
Acknowledgments
We thank Tom Doan, Jan Magnus and Jonas Arias for helpful discussions.
References
1. Inoue, A., Kilian, L., 2013. Inference on impulse response functions in
structural VAR models. Journal of Econometrics 177, 1-13.
2. León, C.A., Massé, J.-C., Rivest, L.P., 2006. A statistical model for
random rotations. Journal of Multivariate Analysis 976, 412-430.
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3. Magnus, J.R., Neudecker, H., 1999. Matrix Differential Calculus with
Applications in Statistics and Econometrics. Wiley, Chichester, UK.
4. Magnus, J.R., 1988. Linear Structures. Oxford University Press, Oxford, UK.
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